IOURNAL
OF ECONOMIC
THEORY
50,
Competitive
Overlapping
155-174
(1990)
Equilibria and the Core of
Generations
Economies
J. ESTEBAN*
lnstituto
de Andlisis
Econcimico,
CSIC
und
Departament
d’Economiu
i Histbria
Econdmica.
Unirersitat
Authoma
de Barcelona.
Beliaterra.
Barcelona.
Spain
AND
T. MILLAN
Departamento
Uniuersidad
Received
de Eeonomia
Complutense,
November
Aplicada,
Madrid,
Spain
28, 1986 and April
5, 1989
This paper analyzes
whether
the well known
result that for Arrow-Debreu
economies
all competitive
equilibria
belong to the core holds true in overlapping
generations
economies.
In these economies competitive
equilibria
are not a subset
of the core. We define the classical set as the set of all those competitive
allocations
which cannot be affected by a monetary
tax-transfer
policy. Then, we demonstrate
that every competitive
equilibrium
in the classical set belongs to the core of the
economy.
Further,
we show that for large economies
no monetary
equilibrium
belongs to the core. Journal
of Economic
Liferature
Classification
Numbers:
021,
023, 111.
c 1990 Academic Press. Inc
1. INTRODUCTION
Overlapping generations models have recently been analyzed from a
game theoretic point of view. Specifically, the core of economies with an
overlapping generations structure has been studied by Hendricks et al.
[15], Kovenock [17], Esteban [9], and Chae [6], showing that competitive equilibria may not belong to the core. Related concepts such as
bounded core and short-run core have been dealt with by Hendricks et al.
[15], Chae [6], Chae and Esteban [7], and Esteban [lo]. In this paper
* I am thankful
for the suggestions
of a referee. Financial
Fundacibn
del Banco Exterior
is gratefully
acknowledged.
support
from
the CAICYT
and
155
0022-053
l/90 $3.00
Copynghi
( I990 by Academic Press. Inc
11, .,“L,, ^r _n__ ._I. ..I.^^ .- . r--
156
ESTEBAN
AND
MILLAN
we demonstrate
that the well known
result that for Arrow-Debreu
economies all competitive equilibria belong to the core may hold true for
overlapping generations economies when we restrict to a specific set of
consumption allocations, the “classical set.”
For finite horizon Arrow-Debreu
economies the relations between
efficiency, competitive equilibria, and core have been established in a series
of well known theorems, as in Arrow and Hahn [l] for instance. On the
one hand, the Fundamental
Welfare Theorems state that competitive
equilibria are Pareto optimal and that Pareto optimal allocations are
implementable as competitive equilibria. On the other hand, we have the
result that every consumption
allocation implementable as a competitive
equilibrium belongs to the core of the economy for all the endowments
allocations for which it can be obtained as a competitive equilibrium.
Samuelson [ 191 himself made it plain that the Fundamental Welfare
Theorems are not valid in overlapping generations economies. However, in
an important
paper Balasko and Shell [3] have demonstrated
that an
equivalence between competitive and efficient allocations can be obtained
by an appropriate redefinition of efficiency. They introduce the notion of
weak Pareto optimality
and demonstrate
that it is satisfied by every
competitive equilibrium. Likewise, Chae’s [6] work can be seen as an
attempt at saving the relation between competitive equilibria and core by
redefining the notion of socially viable allocations. He proposes the concept
of the bounded core and shows that the competitive equilibria of the
economy belong to the bounded core.
In this paper we examine the relation between the notions of Pareto
optimality, competitive equilibria, and core. At variance with the works
reported above, our approach does not consist in redefining the concepts
of dynamic efficiency and/or core. Instead, we shall establish the relations
between the standard concepts but restricted to the “classical set”. The
classical set, %‘, consists of those competitive equilibrium allocations which
cannot be affected by monetary tax-transfers policies, as studied by Balasko
and Shell [4, 51. Specifically our results are the following: (i) the classical
set is a subset of the set of Pareto optimal allocations; and (ii) every competitive equilibrium in the clasical set belongs to the core. The fact that
competitive equilibria not in the classical set may not be Pareto optimal or
not belong to the core suggests that the fundamental singularity of overlapping generations models lies in the possibility of competitive equilibria in
which consumers violate their budgets in every period.
Let us now briefly discuss the relation between these results and the
existing literature. For the case of exchange economies with one agent per
generation, Hendricks et al. [ 151 and Esteban [9] have demonstrated that
all Pareto optimal Walrasian equilibria belong to the core. We know that
in overlapping generations economies Walrasian equilibria might not be
OVERLAPPING
GENERATIONS
157
ECONOMIES
Pareto optimal and thus cannot belong to the core. It can be tempting to
conclude, as suggested by Hendricks et al. [S], that the only reason why
Walrasian equilibria may not belong to the core is that they may fail to be
dynamically efficient. That this is not the case for economies with many
agents per generation has been made plain by Kovenock [ 171 through an
example of a Pareto optimal Walrasian equilibrium which does not belong
to the core, and the core being empty. In this respect, we give here another
example of a Pareto optimal Walrasian allocation which does not belong
to the core, while the core is not empty, and show that these examples are
by no means pathological. We demonstrate in Proposition 3 that for every
competitive consumption
allocation with a price sequence such that
Lim inf, - ,~ llp’ll 3 E > 0 one can construct a distribution of initial endowments for which that equilibrium consumption allocation is not in the core.
Moreover, we prove in Proposition 2 that Lim inf, _ % ilp’ll = 0 is a sufficient condition for a competitive equilibrium to belong to the core of the
economy, irrespective of the distribution
of endowments.]
This result is
stronger than Chae’s [6] Theorem 4.1, where with a continuum of agents
he finds that a sufficient condition for an allocation to belong to the core
is that the present value of the total endowment be finite.
Besides the eventual interest of these results for general equilibrium
theory, the propositions presented in this paper have a special bearing on
monetary theory. Specifically, monetary and IOU equilibria can never
satisfy the sufficiency condition on prices for a competitive equilibrium to
belong to the core. Moreover, we show that every monetary equilibrium
becomes excluded from the core upon replication of the economy. These
results are rather negative concerning the suitability of overlapping generations models for the analysis of fiat money. In our discussion at the end of
the paper we suggest that our results can be interpreted as a formal
demonstration
of Clower’s
[S] observation that, as quoted in de Vries
Il.2113“money differs from other commodities in being universally acceptable as an exchange intermediary by virtue not of individual choice but
rather by virtue of social contrivance” (pp. 14-15).
The paper is structured as follows. Section 2 contains the description of
the model and the basic assumptions. In Section 3 we define and characterize the classical set. Section 4 provides an example of a Walrasian Pareto
optimal equilibrium which does not belong to the core, while the core is
not empty. The relation between efficiency, competitive equilibria, and the
core of the economy is the object of Section 5. There we state our main
propositions.
Section 6 focusses on monetary equilibria and we prove that
monetary equilibria do not belong to the core of large economies. The
paper ends with a discussion of the implications of our propositions
for
’ The
fiber
structure
of the equilibrium
set 1s crucial
for these results;
see Balasko
[2]
158
ESTEBAN AND MILLAN
monetary theory. We examine the relation with the work of Douglas Gale
[12] on the trustworthiness of intertemporal allocations and the role of
money. We argue that our results provide a rationale for the lack of trust
in the IOU competitive equilibria.
2. NOTATION,
ASSUMPTIONS,
AND DEFINITIONS
We shall assume a pure exchange economy with 1 perishable commodities available at every date. In every period t, t = 1, 2, .... a number n
of agents are born’ and live for two periods. At the beginning of this
economy there exists a generation previously born at time t = 0. Let xi.: ‘.’
be the consumption of good i (i= 1,2, .... /) at period t + s (s = 0, 1)
(t= 1, 2, ...) by consumer j (j= 1, 2, .... n) born at t (t=O, 1, 2, ...).
x:.)” E lx’+ be the consumption vector at period t + s (s = 0, 1) by agent j
born at t; x,. Jo rWf the vector of consumptions corresponding to the two
periods that agent will be alive; X,‘+’ E lR’+ be the vector of consumptions
by generation f at period t + s; and X, E rWy the vector of consumptions of
generation t. For generation t =0 we have that .Y~=.x~ and, of course,
X,ER’+. We shall use .Y to denote the sequence x = (.x0, or, .x2, ...I.
Similarly, we shall denote by CJJ:.:“~’the endowment of good i at t+s by
agent j born at period I, by o:.:’ E R’+ the endowment vector at period
t + s (s = 0, 1) of agent ,j born at t, by o,., E Cwy the vector of endowments
corresponding to the two periods that agent ,j will be alive, by w:+.‘E R’+
the vector of endowments of generation t at period t + s, and by w, E [w:
the vector of endowments of generation t. For generation t =0 we have
that w,=oA and ~,ER’+. Finally, we shall denote by w the sequence
w= {coo, co,, Co*,... . . Let Q be the set of all sequences o which are
uniformly bounded from above.
Let us denote by 0’ the aggregate endowments available at period t and
by 0 the sequenceW = 10’ o’, ... ). Since w EQ, it is obvious that WE 1;2as
well. Given a sequence of’aggregate endowments W, we shall denote by
W(W) the set of sequencesof individual initial endowments such that
f
s=O,l
and t=1,2 ,....
w,~,+o:=w,.
O:m.sEw+,
Preferences of consumer j born at t can be represented by a utility
function u,,,: rWy + R, for t = 1, 2, .... and uO,,: R’+ 4 R, for t = 0, with
j = 1, ...) II.
Assumption 1. u,,., has strictly positive first order partial derivatives and
is strictly quasi-concave.
2 The model can be trivially
generalized
to a variable
uniformly
bounded
above by a finite number II.
number
of agentspergeneration
n(t)
OVERLAPPING
DEFINITION
GENERATIONS ECONOMIES
159
1. A consumption allocation sequencex is feasible if
.y:- , +x~=x’6w~~,
for
+o;=co’
t = 1, 2, ... .
DEFINITION 2. The feasible consumption allocation sequence x is
Pareto optimal if there is no feasible 1 such that u,,~(-?~,.,)3 u,. ;(.x,, ,) for
t = 0, 1, 2, and j = 1, .... II with at least one strict inequality.
DEFINITION 3. The feasible consumption allocation sequence .Y is
weakly Pareto optimal if there is no feasible -q, with .Zr,i = x,.,, except for a
finite number of periods, and such that u,,,(Z,, i) 3 u,,,(x,. j) for t = 0, 1, 2, ...
and ,j= 1, .... n, with at least one strict inequality.
Full information and perfect foresight are assumedthroughout the paper.
Let p’,’ denote the price of commodity i delivered at period t, i = 1, .... I and
t = 1, 2, .... J/E DP+ the vector of prices at t, and p the price sequence
p = {p’, p’, j. We shall normalize prices by setting p’.’ = 1 and denote by
g the set of such price sequences,i.e., 9 = (p/p’.’ = 1, p’ E OX’+1..
Following Balasko and Shell’s [4, 51 we shall now introduce the notion
of a price-income equilibrium.
DEFINITION 4. Let I be a consumption allocation sequence. We shall
say that the price sequence p~9
supports s if we have that
u,, ;(.Y,,,) 3 u,, ;(-?,, ,) for all I,. , satisfying
p’..~:I,+p’+‘.-u::‘3p’.I:[,+p’+‘.l:,:
for j= 1, .... n and
and for
t
= 0, u,,, (.Y~i) 3 uo,,(-To,,) for all
.fo.
t = 1, 2, ....
j Satisfying
p’ ..K;, j3p1 ..th j, j= 1, .... n.
Further, for the sequence M’= {M’~, M”, u’~, ... ), MJ,ER:, we shall say that
(p, ~1) is a price-income equilibrium if p supports x and if u’ satisfies
pr . x:f, +p’+ ’ .x:.: ’ = M’,,~for j = 1, .... n and t = 1, 2, .... and p1 .xA , = wo,;
forj= 1, .... n.
We shall now introduce the notion of competitive equilibrium.
DEFINITION 5. A competitive equilibrium is a sequence of commodity
prices p~9’, vector p=(pl, .... ,u,,), PEE”, xj Ip,) = IcjpjI, and endowments allocation sequenceo ESz, and a consumption allocation sequence.Y
such that:
160
ESTEBAN
AND
MILLAN
(i) pf-.~:fj+p’+’
..~i,f’ d p’.e~i,~+p’+’
.o:,f’
and u~,~(x,.,) 3
u,.~(?~.~) for all 2, satisfying the budget constraint,
for t = 1, 2, .... and
’ < ’ .@I~,,
j= 1, ...) n, and for t=O p I ..xO,,,p
’ + P, and ~0, f(xo. ,13 uo, j-z, i) for
all ,Yo,i satisfying the budget constraint; and
(ii)
A monetarql
X-,
+ si = ~0~~~ + 0: for t = 1, 2, ... .
equilibrium
is a competitive equilibrium where p > 0.
Throughout the paper we shall make extensive use of Balasko and Shell’s
[3] characterization of Pareto optimum allocations.3 Therefore, we shall
assumethat the conditions given in their Theorem 5.3 are satisfied.
Assumption
2. (a) The Gaussian curvature of consumer (t, j)‘s indifference surface through x,, ,, 0 < x,, j < [xl, .Y’+ ‘1, is uniformly bounded
above and below away from zero;
(b) There exists a constant Z7, independent of t such that the supporting prices of x satisfy
0 < n<p’~i/llp’ll,
for
i=
1, .... I and
t = 1, 2, ....
(c) The sequence s is uniformly bounded above and below by a
strictly positive vector.
We shall borrow from Esteban [9] the definitions of coalition and core
of the economy and adapt them to the case of many consumers per generation.
DEFINITION
6.
A coalition
is a non-empty connected subset S of the set
of all agents.
We shall denote by S, the set of all agents born at period t which belong
to coalition S, so that S, c S. Then by the conectednessof S we mean that
if S,#Qr and S,+,#@,
then S,+,#Qr, r=l,...,k-I.
A coalition will
thus be formed by a chain of generations, which might not include all
their members. We shall denote by f the first of such generations, i.e.,
f=min{t/S,#@}.
DEFINITION
7. An allocation X is blocked
another allocation x such that:
3 As proven by MillLn
[ 181, Balasko
allocations
can be extended
to economies
and Shell’s
with many
by coalifion
[3] characterization
agents.
S if there exists
of Pareto
optimal
OVERLAPPING
GENERATIONS
161
ECONOMIES
(i) x is feasible for coalition S, i.e.,
(ii) ul., (-“z,j) 3 ur,, CT,.j) f or all agents (t, j) E S with at least one strict
inequality for some agent (t, j) E S,.
DEFINITION
8. The core of an economy is the set of all allocations
are feasible and not blocked by any coalition.
3. THE CLASSICAL SET: DEFINITION
that
AND CHARACTERIZATION
We shall now formally define and characterize the classical set. In order
to do so we shall need the notion of consistency as introduced in Balasko
and Shell [4, 5].5
DEFINITION
9. We shall say that the endowments allocation sequence
o ED is consistent with the price-income equilibrium (p, w) associatedwith
the consumption allocation sequencex if
p’.s:f,+p’+I..~u:,)‘=p’.o:,,+p’+’.w:,)’=~,’,
for
t = 1, 2, ....
and j = 1, .... 12.
We shall denote by Q(x) the set of sequenceso consistent with the
price-income equilibrium associated with .Y. Let us now introduce the
notion of sequential consistency, in which the single condition of life-time
income consistency is replaced by a sequenceof consistency conditions, one
for every period. More precisely, we require this condition to be satisfied
in the long run.
Let us define h:,j= max{p’. [xi, -0:,~],0).,
j=l,...,
n and t=1,2 ,...;
/~:=Cbi,~; and Bi=max{h:,,,
.... e,,, .... hi.,).
i
DEFINITION
10. We shall say that OEQ is sequentially consistent with
the price-income equilibrium (p, ~1) associated with x if o EQ(x) and
Lim inf, _ r B: = 0.
4 Note
that S,- I = 0 and hence for the first generation
the feasibility
condition
reads
c -$,= c 4,
,E \‘,
I t s,
‘For
the relationship
between
equilibrium
and the fiber structure
the notion of consistency
with respect to a price
of the equilibrium
manifold,
see Balasko [2].
income
162
ESTEBAN AND MILLAN
We shall denote by Q”(X) the set of endowment allocations w sequentially consistent with the price-income equilibrium associated with x.
Obviously Q’(x) s Q(X). As we shall see,there are consumption allocations
such that Qs(x) = Q(X), that is, that can only be implemented as equilibria
with the property that in the long run every agent balances his budget constraint in every period of his life. Following Balasko and Shell [4, 53, we
can interpret this property as implying that the government cannot alter
the equilibrium allocation by means of monetary taxes and/or transfers,
i.e., there are no bona fide monetary policies. It is in this sense that we
borrow from David Gale [ 111 the word classical to denote the set of equilibrium allocations in which there is no room for monetary policies. In
other words, allocations in the classical set cannot be competitive equilibria
with either inside or outside money.
We are now in a position to define the classical set.
11. We shall say that the consumption allocation sequence
x belongs to the classical set, XE %, if and only if Q”(x) = Q(X), i.e., all
w E Q(x) are sequentially consistent with the corresponding price-income
equilibrium (p, MI).
DEFINITION
The following proposition provides a characterization of the consumption
allocations sequencesin the classical set by their supporting prices.
PROPOSITION 1. Let Assumptions 1 and 2 be satisfied. The consumption
allocation sequencex belongs to the classical set, .YE %, if and only if its
sequenceof supporting prices satisfies that
Lim inf llp’ll = 0.
f- x
Proof: We shall first show that Lim inf,, r Ilp’ll = 0 implies that
Lim inf, _ m B: = 0.
Let us denote by the superscript k the agent with the largest amount of
borrowing in each generation, i.e., B: =p’ . [xi,, -co:,,].
It is obvious that
11
P’ll . II& -o:.,, II 2 P’ . C& -o&l
= B:.
By Assumption 2(c) the value of I~x:,~- wi,J is uniformly bounded above
by some KC + co. Thus we can write the inequalities
llp’ll K3 /Ip’ll . II&
- o:,~II >p’ . [x;,, -u&l
= B; 2 0.
Therefore when Lim inf, _ ,r llp’ll = 0 it must be that Lim inf,, I Bi = 0.
We shall now demonstrate that when Lim inf,, ,x llp’ll 2 ,?> 0 there
OVERLAPPING
GENERATIONS ECONOMIES
exists o EQ(x) such that Lim inf,, I;o B: > /3 > 0. For that we can
follow the same steps as in Proposition 4 in Esteban [9]. Consider
instance the endowments allocation sequence o and the sequence of
numbers h = (b,, b,, .... b,, ...> such that w,., = .x,.i j= 3, 4, .... n
t = 0, 1, 2, ... and o,, , and o,,~ satisfying
P” c-~:.,-w:.,
I = b, = -p’
163
just
for
real
and
Ix:,, - co:,,1
=-p’+l.[.u::‘-w:f’]=p’+1’[-u:,~‘-o:,:’].
Observe that h, = B:. By Proposition 4 in Esteban [9] the sequence b
exists and is uniformly bounded below by some /I > 0. Further, the endowments allocation sequence thus constructed is such that w E sZ(.u) and hence
.X is a competitive equilibrium.
Q.E.D.
4. COMPETITIVE
EQUILIBRIA
AND THE CORE: AN EXAMPLE
We shall start by giving an example of an economy in which a Pareto
optimal competitive equilibrium
without outside, fiat money does not
belong to the core of the economy, while the set of core allocations is
non-empty.6
Consider a one-good economy with two agents a and b per period
with identical preferences u(.x~, x:+‘) = [x:x:+‘]“’
for t = 1, 2, ... and
’ “’ for agents of generation t = 0. Agents have endowments
44, = c-%I
w I,c,= (1.5, 0.5) and w~,~= (0.5, 1.5) for t = 1, 2, ... and 0,,,=0.5
and
CO~,~= 1.5 for t = 0. See Fig. 1.
The consumption allocation x:[<, = ,~:f; I = .ui b = x:.$ ’ = 1 is a Walrasian
equilibrium
with equilibrium
price sequence ‘p’= 1, for t = I, 2, .... and
x:,~ = 0.5 and .& = 1.5. This Walrasian allocation is Pareto optimal
because the sum of the inverse of prices diverges, thus satisfying Balasko
and Shell’s criterion. This allocation, however, does not belong to the core
because it would be blocked by the coalition formed by agents (t, a)
allocation ,Y, ,( 1.5, 1) and T,.u = (1, 1) for
t = 1, 2, ... with consumption
t = 2, 3, .... Note that the original consumption allocation is a Walrasian
equilibrium with inside money with agents of type “a” behaving as lenders
and those of type “6” as borrowers. The blocking coalition is formed by the
sequence of lenders who prefer to substitute outside money for inside
money.
b Kovenock [17] has produced another example of a Pareto optimum Walrasian equilibrium not in the core, where the set of core allocations is empty. In his example the two
agents of any generation have preferences defined on difIerent goods. Only the two members
of generation I = 0 share their preferences for one common good.
164
ESTEBAN AND MILLAN
agent b
c
t+l
’ t,a
1
0.5
t+l
’ t,b
-e
agent a
1.5 ‘;.a
1
FIGURE
r
1
The set of core allocations is not empty. Consider for instance the weakly
Pareto optimal allocation %A.,= 0.5 and &, = 1.5, -7c,,,= (1.1, 1.1) and
Z,,, = (0.9,0.9), t = 1, 2, . This allocation has supporting prices p’ = 1 and
is thus Pareto optimal. It is a matter of routine to check that this allocation
cannot be blocked. Any agent heading a blocking coalition needs a strictly
positive compensation in the subsequent period. But no matter the type of
agent he tries to get into the coalition, the sequence of compensations is
strictly increasing and eventually becomes unfeasible.
5. EFFICIENCY, CORE, AND COMPETITIVE
It is quite
librium, i.e.,
belong to the
non-autarkic
EQUILIBRIA
obvious that any autarkic Pareto optimal competitive equione in which x,,, = o,,~ for j= 1, .... n and f = 1, 2, .... will
core of the economy. We shall study the relationship between
competitive equilibria and the core.
2. Let Assumption 1 be satisfied. Let x be a consumption
sequence and p its supporting price sequence. Thtk if
Lim inf, _ r llp’ll = 0 the consumption allocation x belongs to the ,core for
eoery consistent endowmentssequencew EQ(x).’
PROPOSITION
allocation
’ Observe that our result is stronger than Chae’s [6] Theorem 4.1, where with a continuum
of agents he finds that a sufficient condition
for an allocation
to belong to the core is that the
present value of the total endowment
be tinite.
OVERLAPPING
GENERATIONS
165
ECONOMIES
Proof:
First note that a price sequence with Lim inf,, r llp’li = 0
cannot correspond to a monetary equilibrium as shown, for instance, in
Esteban [IS]. Let us now assume that x is blocked by coalition S with
consumption allocation 1. Without loss of generality we shall assumethat
~,;~(-f,,,) > z+,(x ) for some (A j) E S,. For any agent (t, j) E S,, T,., must
not be dispreferrzh to x 1,,, so that
p*. [z;,;-x;,,]
+p’+ l. [a:~;’ -x:.;
‘130,
for ail (t, j) E S.
11)
Since by assumption 2 is strictly prefered to x by some member of generation A we have that condition (1) holds as a strict inequality for some
agent (,J j) ES,. From the individual budget constraints we have that
p’&+pf+’
.x;,;‘=pt.w::,+p’+’
.co;,y,
for all (t, j).
Substituting in (1) we obtain
p’.[~~,i-w~,;]+p~+‘-[I~,~‘-o~~~‘]~o
for all (t, j)E S, with strict inequality for some (f, j)ESf.
(t, j)ES,, we have
p’. c [a;.j-w;.,]
I t .s,
+p’+‘.
(2)
Adding over all
jL CT-:,;’ - 4,: ‘13 0
(3)
for t >f, with strict inequality for t =f: Consumption allocation 2 must be
feasible for the coalition members; that is,
c
II-c
]+
I.;-+‘.,
I E s, I
c
for
[.~:,;-w:,,]=o
t2.f:
(4)
/ES,
Combining (3) and (4) we obtain
P‘+I.
c [.qf’-w;,)‘]>p’.
,Es,
c [UC; l.,-co;P,,j]>
/tS,+~
>p’+‘.
C [.T~~‘-w{~‘],
ItsI
...
for
>f+
1.
(5)
We shall now show that
p”+ ’ . c [.f,f; l -co;;
IEs,
‘1 > 0.
Iff= 0, this follows from the fact that 2;. , must be strictly preferred to xh,
for some (0, j) E S,. If f> 0 the feasibility condition (4) imposes that
c [if{,-o$,]
/ E.s/
=o.
166
ESTEBAN AND MILLAN
Thus, from (3) and bearing in mind that Z,,, must be strictly preferred to
x~,, the above inequality follows. The terms of the sequenceof inequalities
(5) can be bounded above by
For economies with w ECJ
is uniformly bounded above.
Hence, if Lim inf,, ~~llp’\l = 0 it must be that
Liminfp’+‘.
I+ %
C [.f:,:‘-CO~,f’]=O.
,E s,
Therefore, there does not exist a sequence I satisfying (1) and (4) and .X
cannot be blocked.
Q.E.D.
We shall now study the circumstances under which a competitive equilibrium would not belong to the core of the economy.
PROPOSITION 3. Let Assumptions 1 and 2 he satkfied. Let 2; be a
tonsumption allocation sequence with supporting prices p satisfying
Lim inf, _ 1 llp’ll 3 E> 0. Then there exists w’ EQ(x) ,for which x is a
competitive equilibrium but does not belong to the core.
Proof: Let us start by pointing out that for any endowments sequence
w ED such that
co;=x’ I
and
,I+’
=.Y:+‘,
t = 0, 1, 2, ...)
(6)
P’ cx:, J-w~,i]+p”‘~[x~,:~-w~,~‘]=o.
j=l
, .... n and
t = 0, 1, 2, ....
(7)
consumption allocation x will be a competitive equilibrium.
In order to prove the proposition we need to provide an example of a
reallocation of initial endowments for which the equilibrium consumption
allocation under consideration does not belong to the core, We shall
consider in the first place the one-commodity case, i.e., I= 1. Moreover, we
shall restrict to reallocations of endowments such that o,.,=x,.,
for
j = 3, 4, ...) n and t=l,2,...,
so that the problem is reduced to finding an
appropriate distribution of endowments between two agents in each
generation, agents 1 and 2.
OVERLAPPING
A clear case of endowment
one satisfying
GENERATIONS
allocation
167
ECONOMIES
for which .Y is not in the core is the
o<.Y:,:‘-Q:,:‘~W:=~.,,
t = 1, 2, ... .
(8)
In that case the coalition formed by the sequence of agents (1, t). t = 1, 2, ...
would block .Y.
We need now to show that there exists an endowment
allocation
sequence satisfying (8) as well as (6) and (7). Using (7) (8) can be rewritten as
Let us construct
the endowment
co ::‘=.~:.:‘-[wl,,-xl,,]/p’+‘,
sequence w satisfying
t=1,2 ,...,
(9) such that
withw;
,-.K;,,>O.
(10)
For every value of 0i.r > x;., and by the use of (10) we can generate a
full sequence w (i.e., w,, 1 and w,.?, t = 1,2, ...) satisfying the equilibrium
conditions (6) and (7).
It remains only to verify whether at least one of such sequencesof
endowments satisfiesw Es2.By assumption 2(c) the sequencex is uniformly
bounded from below by some i >O. It is easy to check that for any
sequenceof endowments obtained from (lo), (6), and (7) such that
C4., -x:,,]/p’<A
for
t = 1, 2, ...
(11)
the consumption allocation x is a Walrasian equilibrium, but does not
belong to the core by construction. It is now immediate that whenever
Lim inf, + x II~‘112 E> 0 there exists some to:,, satisfying (11) such that
0; i xx: I. This completes the proof for n = 1.
The extension to the many commodities case is quite straightforward.
From Assumption 2(b) we have that whenever Lim inf, _ ~ llp’ll > E > 0,
we have that Lim inf, _ ~ p’,’ 3 17. E> 0, i = 1, .... 1.Therefore we can choose
0,. i = x,,, for j= 3, 4, .... n and t = 1, 2, ... and CI~:,:‘.’ = .x:,: ‘.’ for r = 1, 2,
s= 1, 2, i=2, 3, .... I, and t= 1, 2, .... Then our result for I= 1 applies.
Q.E.D.
Proposition 3 can be seen as a source of examples of Pareto optimum
competitive equilibria that do not belong to the core of the economy. The
example provided in Section 2 of a Pareto optimal competitive equilibrium
not in the core is not exceptional.
We have already pointed out in the introduction that for Arrow-Debreu
economies the classical set coincides with the Pareto set. Thus the standard
168
ESTEBAN AND MILLAN
results relating competitive equilibria, efficiency, and core can be restated
in terms of the classical set. Specifically, we can say that in Arrow-Debreu
economies (i) all the allocations belonging to the classical set belong to the
Pareto set (in fact the two sets coincide) and (ii) all the allocations
belonging to the classical set belong to the core for all the endowment
allocations for which they can be implemented as competitive equilibria.
We shall now show that these two propositions hold true for overlapping
generations economies once the analysis is restricted to the classical set, i.e.,
to the set of allocations for which there is no bona tide tax-transfer policy
as studied by Balasco and Shell [4, 51.
PROPOSITION 4.
sumption allocation
Pareto optimal.
Let Assumptions
1 and 2 he satisfied. Let x be a consequence belonging
to the classical set, x E 97. Then x is
Proof:
This follows immediately from Proposition 1, taking into
account Balasko and Shell’s [3] proposition that if the supporting prices
of a weakly Pareto optimal consumption allocation satisfy that
Lim inf, _ x llp’ll = 0 then that consumption allocation is Pareto optimal.
Q.E.D.
PROPOSITION 5. Let Assumptions
1 and 2 be satisfied. Then the consumption allocation
sequence x belongs to the core for every w E Q(x) if and only
if it belongs to the classical set, I E %7.
Proof.
Proposition 1 establishes that the supporting prices of a consumption allocation sequence satisfy Lim inf, _ ~, IIp’(I = 0 if and only if x
belongs to the classical set. Thus, Propositions 1 and 2 together imply the
sufftciency part of this proposition; that is, if a consumption allocation
belongs to the classical set it belongs to the core for all endowment sequences for which it is competitive equilibrium. Further, Propositions 1 and 3
together imply the necessity part of Proposition 5. Taken together, they say
that if a consumption allocation does not belong to the classical set there
exists an endowment allocation for which that consumption allocation does
Q.E.D.
not belong to the core while still being a competitive equilibrium.
6. MONETARY
EQUILIBRIA
AND THE CORE
We have already pointed out that overlapping generations models have
been considered as the most appropriate framework for the analysis of fiat
money. It is thus natural to pay special attention to the relationship
between monetary equilibria and core allocations. Besides the obvious
relevance of our previous results to monetary equilibria, we shall now
OVERLAPPING
169
GENERATIONS ECONOMIES
introduce some additional results specifically referring to monetary allocations. We start by demonstrating that monetary equilibria with too much
money will not belong to the core. But the main result of this section is
that as we enlarge the economy by replication every monetary equilibrium
eventually becomesexcluded from the core.
With many consumers per generation we may have IOU equilibria,
monetary equilibria, and a mixture of the two, i.e., simultaneous use of
IOUs and fiat money. The following result refers to economies in which in
the long run all intertemporal purchases tend to be made for fiat money.
This is an extreme case of a competitive equilibrium in which not only is
fiat money the only means of transferring purchasing power from present
to the future, but money is present in all transactions. We show that these
equilibria do not belong to the core.
6. Let Assumption 1 be satisfied. Let x be a competitive
equilibrium consumption allocation sequenceand p the sequence of equilibrium prices. Then if
PROPOSITION
Liminfp’+‘.Ix:+‘-w:+‘I/~=l,
r + ,-c
P>O>
(12)
the consumption allocation x does not belong to the core.
Proof. From Definition 5 we know that p’+ ’ [xi+ ’ -CO:+ ‘I= p =
xj p.i’ t = 0, 1, 2, ... . Therefore, if (12) is satisfied we have that
Liminfp’+‘.[.ti+‘-wi+‘]/p’+‘.[.u:+’-o:+’]=l.
f - .%
Thus it must be that
Lim inf p’ + ’ . [x;,:‘-u;,;‘]
,4 x
=Liminfp’+’
, 4 %I
. [x:,; ’ -co;.; ‘1 = z, 3; > 0,
j=l
7---,n,
where ZE R’+ and z(i)= min{z,(i), .... z,(i), .... z,(i)), i= 1, .... 1.
Hence there exist k and 1, 0 < ,462 such that [xi,:’ - wi.f ‘I> 1 for
j=l,...,
mand r=k,k+l,k+2
,..,.
Consider now coalition S formed by agents (t, j), j= 1, .... n and
t=k+ 1, k+2, ... with consumption allocation X such that
x,, j = x, j
forall (t,j)ES,,
t=k+2,k+3
Jyk+ ,, j = C-Y;1:. , + I., x”, 1 f. ,I,
,...
j = 1, .... n.
and
170
ESTEBAN
AND
MILLAN
Allocation .Y is feasible for coalition S, is as good as .Y for all members of
S, and is strictly preferred by all members of S, + 1. Therefore, consumption
Q.E.D.
allocation x does not belong to the core.
This result seems to substantiate the interpretation
given in Esteban [7]
to the effect that it is acting as a means of exchange rather than as a store
of value that confers social acceptability on fiat money. We have a case in
which in the long run all exchanges are intertemporal and made by means
of money. Thus, the proposition
that no such equilibrium belongs to the
core reinforces that view in a many agents economy.
In the following
proposition
we show that for large economies no
monetary equilibrium belongs to the core.
PROPOSITION 7. Let < he an economy with m agents and n goods in u>hich
Assumptions 1 and 2(a) are satisfied. Let x he a monetary equilibrium consumption allocation of the economli 5 and p the equilibrium price sequence.
Then there exists K such that for the Kth replica of the econom.v t(K), the
consumption allocation c(K) does not belong to the core.
Proof: Consider the (k + 1) replica of the economy and let S be a coalition formed by all agents (t, j), j= 1, .... n, t = 1, 2, .... in e(k + 1) and all
agents (0, j), j = 1, .... n, in t(k). Since I is a monetary equilibrium we have
that p’ . [o: - x:] = A4 > 0. Therefore, there exists at least one vector
2; jE IF!/+ such that pl. [.?i,i-xl,j]
>O, j= 1, .... n and .Y; --w; 60. Consider now the consumption allocation X for the members of coalition S
such that x[,, = x’,.~ for all (t, j) E S,, t = 0, 2, 3, ... and .Y,,, = [(.?I., + k.xi. ,)/
(k+ l), XT,,] for all (1, j)ES!.
Observe that .Uis obtained as a linear convex combination of vectors .Y~,,
and .?,,j= [.?l,,, X’,,, ] and that as k becomes large “I. j -+x,,,. Thus, we
have that
-I
p I .x,,+p2..r;.,>p’
.“:,;+p2.xf~,’
j = 1, .._,fz.
Moreover, it is easy to check that consumption allocation X is feasible.
By Assumptions 1 and 2(a) there exists a finite K such that
u,,,[(M<~~j+K.x~,j)/(K+
I),
xf.iI >Ul.,[x;.j’ -‘f.jI,
j=l
, .... 12.
See Figure 2. Since X,,j = _Y,.~ for the rest of the members of S, S will block
consumption allocation x.
Q.E.D.
This proposition can be interpreted as a formal proof of the observation
by Clower that money is held not as the result of individual voluntary
choice but by social contrivance, as pointed out in the Introduction.
171
oVERLAPPINGGENERATIoNSECONOMIES
7. FINAL REMARKS ON SOME To~rcs IN MONETARY
THEORY
We have just demonstrated that for large economies no monetary equilibrium belongs to the core. This result appears to be quite negative with
respect to the important literature on fiat money in overlapping generations economies, as developed in Grandmont [14], Kareken and Wallace
[16], and Sargent [20], among others. However, it can be interpreted in
a more positive spirit as providing a rigorous demonstration of the claim
made by Clower [S] that the social acceptance of money is not voluntary
and based on its virtue of being a store of value. As a matter of fact, there
is nothing terrible or new in this view of money. Douglas Gale [12] and
[ 131, when examining the role of money in the social acceptance of allocations. points out that “it is not the invention of paper money which restores
trustworthiness. The Walras allocations are trustworthy in the monetary
economy only because there is, in the background, a government which
can enforce, evidently at no cost, the payment of [money] taxes. Thus, we
have introduced not just a new commodity (money) but a new social
institution” (p. 465). From this point of view it is obvious that fiat money
has been introduced in overlapping generations models as a commodity
and not as a social institution.8
Let us be more specific in comparing our results with Douglas Gale’s
[12] work. A we have already pointed out, he has shown that the intro’ In this respect, de Vries [21] has recently examined
the case in which the acceptance
fiat money is made compulsory.
i.e., money is given the status of “legal tender.”
of
172
ESTEBAN
AND
MILLAN
duction of money may help in making socially acceptable allocations which
would have been blocked without its help. In Gale’s model there is a finite
number of periods and, if being an Arrow-Debreu
economy in every
respect, competitive equilibria belong to its core. However, Gale argues
that in those equilibria in which there is net borrowing and lending, lenders
have good reasons not to trust borrowers.
It is in their interest to break
futures contracts in later periods of their lifes. Gale thus defines the concept
of sequential core, i.e., those allocations which belong to the core both in
the lirst and in the subsequent periods, and shows that an allocation is
trustworthy
if and only if it belongs to the sequential core. Thus, lenders
would not trust allocations out of the sequential core even if they belong
to the core of the economy. In order to circumvent
this problem
he proposes that the social institution
of money may render otherwise
untrustworthy
allocations worthy of trust. As in all finite horizon monetary
models he needs to introduce a money tax at the end in order to make
money valuable. He then finds that in this economy competitive monetary
equilibria do belong to the sequential core.
From this point of view, our results can be interpreted in the following
way. It seems natural to reconsider his problem in an overlapping generations economy in which one does not need the device of the money tax in
order to make money valuable. Then, our findings that Pareto optimal
monetary competitive equilibria might not belong to the core and that
when the economy becomes large by replication no monetary allocation
belongs to the core seem to confirm Gale’s assertion that what makes
allocations socially stable in his model is not the introduction
of fiat
money, but the making its use compulsory.
We can go deeper in comparing our model with Douglas Gale’s. As we
shall now argue, our results can be seen as providing a rationale for Gale’s
assumption on agents not honoring their contracts. Let us start by noting
that an IOU equilibrium in an overlapping generations model can be
understood as a sequence of overlapping finite horizon Gale’s competitive
equilibria. Further, observe that in those equilibria contracts in the futures
markets are signed on the two sides by consumers belonging to the same
generation. Therefore, as far as futures markets are concerned, IOb equilibria can be seen as a sequence of isolated generations, i.e., as a sequence
of overlapping two-period Gale’s equilibria. Alternatively, Gale’s model can
be considered as isolating one single generation of an IOU equilibrium
sequence from an overlapping generations model in order to examine its
behavior in the futures market.
In spite of their similarity, the two models seem to yield different results.
While in Gale’s model all competitive equilibria belong to the core, in our
model their equivalent, i.e., our IOU equilibria, do not. Thus, the fact of
placing a collection of self-contained economies one after the other breaks
OVERLAPPING
GENERATIONSECONOMIES
173
the relation between competitive equilibria and core. In other words, from
a game-theoretic point of view it makes a substantial difference whether we
consider an isolated finite chain of periods or the full infinite sequence.As
we have seen in Proposition 5, only those Walrasian equilibria in which
there is no borrowing and lending in the long run are always in the core.
Hence, the core of an overlapping generations economy formed by chaining a sequence of Gale’s two period economies would not contain the
allocations that do not belong to the sequential core in Gale’s model.
Therefore, one need not suppose that agents do not honor their contracts
to claim that equilibria which involve borrowing might not be viable. The
mere fact that agents live in an endlesschain of generations can make IOU
equilibria untrustworthy. Our results can thus be considered as a rationale
for using the concept of sequential core in finite horizon economies.
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